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Theorem rdgeqoa 37353
Description: If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)
Assertion
Ref Expression
rdgeqoa ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))

Proof of Theorem rdgeqoa
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . 2 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → 𝑋 ∈ ω)
2 eleq1 2827 . . . . 5 (𝑥 = 𝑋 → (𝑥 ∈ ω ↔ 𝑋 ∈ ω))
323anbi3d 1441 . . . 4 (𝑥 = 𝑋 → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω)))
4 oveq2 7439 . . . . . . 7 (𝑥 = 𝑋 → (𝑁 +o 𝑥) = (𝑁 +o 𝑋))
54fveq2d 6911 . . . . . 6 (𝑥 = 𝑋 → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)))
6 oveq2 7439 . . . . . . 7 (𝑥 = 𝑋 → (𝑀 +o 𝑥) = (𝑀 +o 𝑋))
76fveq2d 6911 . . . . . 6 (𝑥 = 𝑋 → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))
85, 7eqeq12d 2751 . . . . 5 (𝑥 = 𝑋 → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))
98imbi2d 340 . . . 4 (𝑥 = 𝑋 → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))))
103, 9imbi12d 344 . . 3 (𝑥 = 𝑋 → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))))
11 peano1 7911 . . . . 5 ∅ ∈ ω
12 oa0 8553 . . . . . . . . . . . 12 (𝑁 ∈ On → (𝑁 +o ∅) = 𝑁)
1312fveq2d 6911 . . . . . . . . . . 11 (𝑁 ∈ On → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐴)‘𝑁))
1413eqcomd 2741 . . . . . . . . . 10 (𝑁 ∈ On → (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
15 oa0 8553 . . . . . . . . . . . 12 (𝑀 ∈ On → (𝑀 +o ∅) = 𝑀)
1615fveq2d 6911 . . . . . . . . . . 11 (𝑀 ∈ On → (rec(𝐹, 𝐵)‘(𝑀 +o ∅)) = (rec(𝐹, 𝐵)‘𝑀))
1716eqcomd 2741 . . . . . . . . . 10 (𝑀 ∈ On → (rec(𝐹, 𝐵)‘𝑀) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
1814, 17eqeqan12d 2749 . . . . . . . . 9 ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
1918biimpd 229 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
20 eleq1 2827 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑥 ∈ ω ↔ ∅ ∈ ω))
21203anbi3d 1441 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω)))
2211biantru 529 . . . . . . . . . . . 12 (𝑀 ∈ On ↔ (𝑀 ∈ On ∧ ∅ ∈ ω))
2322anbi2i 623 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
24 3anass 1094 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω) ↔ (𝑁 ∈ On ∧ (𝑀 ∈ On ∧ ∅ ∈ ω)))
2523, 24bitr4i 278 . . . . . . . . . 10 ((𝑁 ∈ On ∧ 𝑀 ∈ On) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ ∅ ∈ ω))
2621, 25bitr4di 289 . . . . . . . . 9 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On)))
27 oveq2 7439 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑁 +o 𝑥) = (𝑁 +o ∅))
2827fveq2d 6911 . . . . . . . . . . 11 (𝑥 = ∅ → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o ∅)))
29 oveq2 7439 . . . . . . . . . . . 12 (𝑥 = ∅ → (𝑀 +o 𝑥) = (𝑀 +o ∅))
3029fveq2d 6911 . . . . . . . . . . 11 (𝑥 = ∅ → (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))
3128, 30eqeq12d 2751 . . . . . . . . . 10 (𝑥 = ∅ → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))
3231imbi2d 340 . . . . . . . . 9 (𝑥 = ∅ → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅)))))
3326, 32imbi12d 344 . . . . . . . 8 (𝑥 = ∅ → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o ∅)) = (rec(𝐹, 𝐵)‘(𝑀 +o ∅))))))
3419, 33mpbiri 258 . . . . . . 7 (𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
3534ax-gen 1792 . . . . . 6 𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
36 sbc6g 3821 . . . . . 6 (∅ ∈ ω → ([∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ∀𝑥(𝑥 = ∅ → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))))
3735, 36mpbiri 258 . . . . 5 (∅ ∈ ω → [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
3811, 37ax-mp 5 . . . 4 [∅ / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
39 peano2b 7904 . . . . 5 (𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
40393anbi3i 1158 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω))
4140imbi1i 349 . . . . . . 7 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
42 nnon 7893 . . . . . . . . . . . . 13 (𝑥 ∈ ω → 𝑥 ∈ On)
43 oacl 8572 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ On ∧ 𝑥 ∈ On) → (𝑁 +o 𝑥) ∈ On)
44 oacl 8572 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ On ∧ 𝑥 ∈ On) → (𝑀 +o 𝑥) ∈ On)
4543, 44anim12i 613 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ On ∧ 𝑥 ∈ On) ∧ (𝑀 ∈ On ∧ 𝑥 ∈ On)) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
46453impdir 1350 . . . . . . . . . . . . . . 15 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) → ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On))
47 rdgsuc 8463 . . . . . . . . . . . . . . . . . 18 ((𝑁 +o 𝑥) ∈ On → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥))))
48 fveq2 6907 . . . . . . . . . . . . . . . . . 18 ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) → (𝐹‘(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥))) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
4947, 48sylan9eqr 2797 . . . . . . . . . . . . . . . . 17 (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ (𝑁 +o 𝑥) ∈ On) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
5049adantrr 717 . . . . . . . . . . . . . . . 16 (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
51 rdgsuc 8463 . . . . . . . . . . . . . . . . 17 ((𝑀 +o 𝑥) ∈ On → (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
5251ad2antll 729 . . . . . . . . . . . . . . . 16 (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)) = (𝐹‘(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
5350, 52eqtr4d 2778 . . . . . . . . . . . . . . 15 (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ ((𝑁 +o 𝑥) ∈ On ∧ (𝑀 +o 𝑥) ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
5446, 53sylan2 593 . . . . . . . . . . . . . 14 (((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ∧ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On)) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
5554ancoms 458 . . . . . . . . . . . . 13 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ On) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
5642, 55syl3anl3 1413 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
57 onasuc 8565 . . . . . . . . . . . . . . 15 ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (𝑁 +o suc 𝑥) = suc (𝑁 +o 𝑥))
5857fveq2d 6911 . . . . . . . . . . . . . 14 ((𝑁 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
59583adant2 1130 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
6059adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐴)‘suc (𝑁 +o 𝑥)))
61 onasuc 8565 . . . . . . . . . . . . . . 15 ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (𝑀 +o suc 𝑥) = suc (𝑀 +o 𝑥))
6261fveq2d 6911 . . . . . . . . . . . . . 14 ((𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
63623adant1 1129 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
6463adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘suc (𝑀 +o 𝑥)))
6556, 60, 643eqtr4d 2785 . . . . . . . . . . 11 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ∧ (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
6665ex 412 . . . . . . . . . 10 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
6766imim2d 57 . . . . . . . . 9 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
6840, 67sylbir 235 . . . . . . . 8 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → (((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
6968a2i 14 . . . . . . 7 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
7041, 69sylbi 217 . . . . . 6 (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
71 sbcimg 3843 . . . . . . 7 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
72 sbc3an 3861 . . . . . . . . 9 ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ ([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω))
73 sbcg 3870 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑁 ∈ On ↔ 𝑁 ∈ On))
74 sbcg 3870 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑀 ∈ On ↔ 𝑀 ∈ On))
75 sbcel1v 3862 . . . . . . . . . . 11 ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω)
7675a1i 11 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]𝑥 ∈ ω ↔ suc 𝑥 ∈ ω))
7773, 74, 763anbi123d 1435 . . . . . . . . 9 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥]𝑁 ∈ On ∧ [suc 𝑥 / 𝑥]𝑀 ∈ On ∧ [suc 𝑥 / 𝑥]𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
7872, 77bitrid 283 . . . . . . . 8 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) ↔ (𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω)))
79 sbcimg 3843 . . . . . . . . 9 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
80 sbcg 3870 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) ↔ (rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀)))
81 sbceqg 4418 . . . . . . . . . . 11 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))
82 csbfv12 6955 . . . . . . . . . . . . 13 suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (suc 𝑥 / 𝑥rec(𝐹, 𝐴)‘suc 𝑥 / 𝑥(𝑁 +o 𝑥))
83 csbconstg 3927 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥rec(𝐹, 𝐴) = rec(𝐹, 𝐴))
84 csbov123 7475 . . . . . . . . . . . . . . 15 suc 𝑥 / 𝑥(𝑁 +o 𝑥) = (suc 𝑥 / 𝑥𝑁suc 𝑥 / 𝑥 +o suc 𝑥 / 𝑥𝑥)
85 csbconstg 3927 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥 +o = +o )
86 csbconstg 3927 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑁 = 𝑁)
87 csbvarg 4440 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑥 = suc 𝑥)
8885, 86, 87oveq123d 7452 . . . . . . . . . . . . . . 15 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥𝑁suc 𝑥 / 𝑥 +o suc 𝑥 / 𝑥𝑥) = (𝑁 +o suc 𝑥))
8984, 88eqtrid 2787 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(𝑁 +o 𝑥) = (𝑁 +o suc 𝑥))
9083, 89fveq12d 6914 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥rec(𝐹, 𝐴)‘suc 𝑥 / 𝑥(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)))
9182, 90eqtrid 2787 . . . . . . . . . . . 12 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)))
92 csbfv12 6955 . . . . . . . . . . . . 13 suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (suc 𝑥 / 𝑥rec(𝐹, 𝐵)‘suc 𝑥 / 𝑥(𝑀 +o 𝑥))
93 csbconstg 3927 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥rec(𝐹, 𝐵) = rec(𝐹, 𝐵))
94 csbov123 7475 . . . . . . . . . . . . . . 15 suc 𝑥 / 𝑥(𝑀 +o 𝑥) = (suc 𝑥 / 𝑥𝑀suc 𝑥 / 𝑥 +o suc 𝑥 / 𝑥𝑥)
95 csbconstg 3927 . . . . . . . . . . . . . . . 16 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥𝑀 = 𝑀)
9685, 95, 87oveq123d 7452 . . . . . . . . . . . . . . 15 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥𝑀suc 𝑥 / 𝑥 +o suc 𝑥 / 𝑥𝑥) = (𝑀 +o suc 𝑥))
9794, 96eqtrid 2787 . . . . . . . . . . . . . 14 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(𝑀 +o 𝑥) = (𝑀 +o suc 𝑥))
9893, 97fveq12d 6914 . . . . . . . . . . . . 13 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥rec(𝐹, 𝐵)‘suc 𝑥 / 𝑥(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
9992, 98eqtrid 2787 . . . . . . . . . . . 12 (suc 𝑥 ∈ ω → suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))
10091, 99eqeq12d 2751 . . . . . . . . . . 11 (suc 𝑥 ∈ ω → (suc 𝑥 / 𝑥(rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = suc 𝑥 / 𝑥(rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
10181, 100bitrd 279 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)) ↔ (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))
10280, 101imbi12d 344 . . . . . . . . 9 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → [suc 𝑥 / 𝑥](rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
10379, 102bitrd 279 . . . . . . . 8 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))) ↔ ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥)))))
10478, 103imbi12d 344 . . . . . . 7 (suc 𝑥 ∈ ω → (([suc 𝑥 / 𝑥](𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → [suc 𝑥 / 𝑥]((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))))
10571, 104bitrd 279 . . . . . 6 (suc 𝑥 ∈ ω → ([suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) ↔ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ suc 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o suc 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o suc 𝑥))))))
10670, 105imbitrrid 246 . . . . 5 (suc 𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
10739, 106sylbi 217 . . . 4 (𝑥 ∈ ω → (((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))) → [suc 𝑥 / 𝑥]((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥))))))
10838, 107findes 7923 . . 3 (𝑥 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑥 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑥)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑥)))))
10910, 108vtoclga 3577 . 2 (𝑋 ∈ ω → ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋)))))
1101, 109mpcom 38 1 ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  [wsbc 3791  csb 3908  c0 4339  Oncon0 6386  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448   +o coa 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509
This theorem is referenced by:  finxpreclem4  37377
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