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Theorem tfsconcatfv2 43330
Description: A latter value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
Hypothesis
Ref Expression
tfsconcat.op + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
Assertion
Ref Expression
tfsconcatfv2 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵𝑋))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧
Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)   𝑋(𝑎,𝑏)

Proof of Theorem tfsconcatfv2
StepHypRef Expression
1 tfsconcat.op . . . . 5 + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
21tfsconcatun 43327 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
32fveq1d 6909 . . 3 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))})‘(𝐶 +o 𝑋)))
43adantr 480 . 2 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))})‘(𝐶 +o 𝑋)))
5 simplll 775 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → 𝐴 Fn 𝐶)
6 simplrl 777 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
7 simplrr 778 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
8 simpr 484 . . . . . . 7 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
9 tfsconcatlem 43326 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
106, 7, 8, 9syl3anc 1370 . . . . . 6 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
1110ralrimiva 3144 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))
12 eqid 2735 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}
1312fnopabg 6706 . . . . 5 (∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
1411, 13sylib 218 . . . 4 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
1514adantr 480 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
16 disjdif 4478 . . . 4 (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅
1716a1i 11 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅)
18 pm3.22 459 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
1918adantl 481 . . . . . 6 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
20 oaordi 8583 . . . . . 6 ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑋𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
2119, 20syl 17 . . . . 5 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑋𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
2221imp 406 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷))
23 simplrl 777 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → 𝐶 ∈ On)
24 simpr 484 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
2524adantl 481 . . . . . 6 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
26 onelon 6411 . . . . . 6 ((𝐷 ∈ On ∧ 𝑋𝐷) → 𝑋 ∈ On)
2725, 26sylan 580 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → 𝑋 ∈ On)
28 oaword1 8589 . . . . 5 ((𝐶 ∈ On ∧ 𝑋 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑋))
2923, 27, 28syl2anc 584 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → 𝐶 ⊆ (𝐶 +o 𝑋))
30 oacl 8572 . . . . . . . . 9 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
31 eloni 6396 . . . . . . . . 9 ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
3230, 31syl 17 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
33 eloni 6396 . . . . . . . . 9 (𝐶 ∈ On → Ord 𝐶)
3433adantr 480 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
3532, 34jca 511 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
3635adantl 481 . . . . . 6 (((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
3736adantr 480 . . . . 5 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
38 ordeldif 43248 . . . . 5 ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑋))))
3937, 38syl 17 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑋))))
4022, 29, 39mpbir2and 713 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
415, 15, 17, 40fvun2d 7003 . 2 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))})‘(𝐶 +o 𝑋)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}‘(𝐶 +o 𝑋)))
42 eqid 2735 . . . . . 6 (𝐶 +o 𝑋) = (𝐶 +o 𝑋)
43 eqid 2735 . . . . . 6 (𝐵𝑋) = (𝐵𝑋)
44 oveq2 7439 . . . . . . . . 9 (𝑧 = 𝑋 → (𝐶 +o 𝑧) = (𝐶 +o 𝑋))
4544eqeq2d 2746 . . . . . . . 8 (𝑧 = 𝑋 → ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)))
46 fveq2 6907 . . . . . . . . 9 (𝑧 = 𝑋 → (𝐵𝑧) = (𝐵𝑋))
4746eqeq2d 2746 . . . . . . . 8 (𝑧 = 𝑋 → ((𝐵𝑋) = (𝐵𝑧) ↔ (𝐵𝑋) = (𝐵𝑋)))
4845, 47anbi12d 632 . . . . . . 7 (𝑧 = 𝑋 → (((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑋) ∧ (𝐵𝑋) = (𝐵𝑋))))
4948rspcev 3622 . . . . . 6 ((𝑋𝐷 ∧ ((𝐶 +o 𝑋) = (𝐶 +o 𝑋) ∧ (𝐵𝑋) = (𝐵𝑋))) → ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))
5042, 43, 49mpanr12 705 . . . . 5 (𝑋𝐷 → ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))
5150adantl 481 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))
52 ovex 7464 . . . . . 6 (𝐶 +o 𝑋) ∈ V
53 fvex 6920 . . . . . 6 (𝐵𝑋) ∈ V
5452, 53pm3.2i 470 . . . . 5 ((𝐶 +o 𝑋) ∈ V ∧ (𝐵𝑋) ∈ V)
55 eleq1 2827 . . . . . . 7 (𝑥 = (𝐶 +o 𝑋) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
56 eqeq1 2739 . . . . . . . . 9 (𝑥 = (𝐶 +o 𝑋) → (𝑥 = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑧)))
5756anbi1d 631 . . . . . . . 8 (𝑥 = (𝐶 +o 𝑋) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))))
5857rexbidv 3177 . . . . . . 7 (𝑥 = (𝐶 +o 𝑋) → (∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))))
5955, 58anbi12d 632 . . . . . 6 (𝑥 = (𝐶 +o 𝑋) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))) ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))))
60 eqeq1 2739 . . . . . . . . 9 (𝑦 = (𝐵𝑋) → (𝑦 = (𝐵𝑧) ↔ (𝐵𝑋) = (𝐵𝑧)))
6160anbi2d 630 . . . . . . . 8 (𝑦 = (𝐵𝑋) → (((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧))))
6261rexbidv 3177 . . . . . . 7 (𝑦 = (𝐵𝑋) → (∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)) ↔ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧))))
6362anbi2d 630 . . . . . 6 (𝑦 = (𝐵𝑋) → (((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧))) ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))))
6459, 63opelopabg 5548 . . . . 5 (((𝐶 +o 𝑋) ∈ V ∧ (𝐵𝑋) ∈ V) → (⟨(𝐶 +o 𝑋), (𝐵𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))))
6554, 64mp1i 13 . . . 4 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (⟨(𝐶 +o 𝑋), (𝐵𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵𝑋) = (𝐵𝑧)))))
6640, 51, 65mpbir2and 713 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ⟨(𝐶 +o 𝑋), (𝐵𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))})
67 fnopfvb 6961 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶) ∧ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵𝑋) ↔ ⟨(𝐶 +o 𝑋), (𝐵𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
6815, 40, 67syl2anc 584 . . 3 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵𝑋) ↔ ⟨(𝐶 +o 𝑋), (𝐵𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}))
6966, 68mpbird 257 . 2 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵𝑋))
704, 41, 693eqtrd 2779 1 ((((𝐴 Fn 𝐶𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ∃!weu 2566  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  cop 4637  {copab 5210  dom cdm 5689  Ord word 6385  Oncon0 6386   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433   +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-pow 5371  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-rmo 3378  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-int 4952  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:  tfsconcatfv  43331
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