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Theorem tfrlem5 7238
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem5 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,,𝑢,𝑣,𝐹   𝐴,𝑔,
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑓)

Proof of Theorem tfrlem5
Dummy variables 𝑧 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 vex 3080 . . 3 𝑔 ∈ V
31, 2tfrlem3a 7235 . 2 (𝑔𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))))
4 vex 3080 . . 3 ∈ V
51, 4tfrlem3a 7235 . 2 (𝐴 ↔ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎))))
6 reeanv 2990 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ↔ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))))
7 simp2ll 1120 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑔 Fn 𝑧)
8 simp3l 1081 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑔𝑢)
9 fnbr 5792 . . . . . . . . 9 ((𝑔 Fn 𝑧𝑥𝑔𝑢) → 𝑥𝑧)
107, 8, 9syl2anc 690 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑧)
11 simp2rl 1122 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fn 𝑤)
12 simp3r 1082 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑣)
13 fnbr 5792 . . . . . . . . 9 (( Fn 𝑤𝑥𝑣) → 𝑥𝑤)
1411, 12, 13syl2anc 690 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥𝑤)
1510, 14elind 3663 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑥 ∈ (𝑧𝑤))
16 onin 5561 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (𝑧𝑤) ∈ On)
17163ad2ant1 1074 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ∈ On)
18 fnfun 5787 . . . . . . . . . 10 (𝑔 Fn 𝑧 → Fun 𝑔)
197, 18syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun 𝑔)
20 inss1 3698 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑧
21 fndm 5789 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
227, 21syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom 𝑔 = 𝑧)
2320, 22syl5sseqr 3521 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom 𝑔)
2419, 23jca 552 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun 𝑔 ∧ (𝑧𝑤) ⊆ dom 𝑔))
25 fnfun 5787 . . . . . . . . . 10 ( Fn 𝑤 → Fun )
2611, 25syl 17 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → Fun )
27 inss2 3699 . . . . . . . . . 10 (𝑧𝑤) ⊆ 𝑤
28 fndm 5789 . . . . . . . . . . 11 ( Fn 𝑤 → dom = 𝑤)
2911, 28syl 17 . . . . . . . . . 10 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → dom = 𝑤)
3027, 29syl5sseqr 3521 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑧𝑤) ⊆ dom )
3126, 30jca 552 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (Fun ∧ (𝑧𝑤) ⊆ dom ))
32 simp2lr 1121 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)))
33 ssralv 3533 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑧 → (∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎))))
3420, 32, 33mpsyl 65 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝐹‘(𝑔𝑎)))
35 simp2rr 1123 . . . . . . . . 9 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))
36 ssralv 3533 . . . . . . . . 9 ((𝑧𝑤) ⊆ 𝑤 → (∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎))))
3727, 35, 36mpsyl 65 . . . . . . . 8 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑎) = (𝐹‘(𝑎)))
3817, 24, 31, 34, 37tfrlem1 7234 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → ∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎))
39 fveq2 5986 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑔𝑎) = (𝑔𝑥))
40 fveq2 5986 . . . . . . . . 9 (𝑎 = 𝑥 → (𝑎) = (𝑥))
4139, 40eqeq12d 2529 . . . . . . . 8 (𝑎 = 𝑥 → ((𝑔𝑎) = (𝑎) ↔ (𝑔𝑥) = (𝑥)))
4241rspcv 3182 . . . . . . 7 (𝑥 ∈ (𝑧𝑤) → (∀𝑎 ∈ (𝑧𝑤)(𝑔𝑎) = (𝑎) → (𝑔𝑥) = (𝑥)))
4315, 38, 42sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = (𝑥))
44 funbrfv 6027 . . . . . . 7 (Fun 𝑔 → (𝑥𝑔𝑢 → (𝑔𝑥) = 𝑢))
4519, 8, 44sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑔𝑥) = 𝑢)
46 funbrfv 6027 . . . . . . 7 (Fun → (𝑥𝑣 → (𝑥) = 𝑣))
4726, 12, 46sylc 62 . . . . . 6 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → (𝑥) = 𝑣)
4843, 45, 473eqtr3d 2556 . . . . 5 (((𝑧 ∈ On ∧ 𝑤 ∈ On) ∧ ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) ∧ (𝑥𝑔𝑢𝑥𝑣)) → 𝑢 = 𝑣)
49483exp 1255 . . . 4 ((𝑧 ∈ On ∧ 𝑤 ∈ On) → (((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣)))
5049rexlimivv 2922 . . 3 (∃𝑧 ∈ On ∃𝑤 ∈ On ((𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
516, 50sylbir 223 . 2 ((∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑎𝑧 (𝑔𝑎) = (𝐹‘(𝑔𝑎))) ∧ ∃𝑤 ∈ On ( Fn 𝑤 ∧ ∀𝑎𝑤 (𝑎) = (𝐹‘(𝑎)))) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
523, 5, 51syl2anb 494 1 ((𝑔𝐴𝐴) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1938  {cab 2500  wral 2800  wrex 2801  cin 3443  wss 3444   class class class wbr 4481  dom cdm 4932  cres 4934  Oncon0 5530  Fun wfun 5683   Fn wfn 5684  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-iota 5653  df-fun 5691  df-fn 5692  df-fv 5697
This theorem is referenced by:  tfrlem7  7241
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