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Theorem cbvfo 6509
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
cbvfo.1 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvfo (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐵(𝑥)

Proof of Theorem cbvfo
StepHypRef Expression
1 fofn 6084 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 cbvfo.1 . . . . . 6 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
32bicomd 213 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜓𝜑))
43eqcoms 2629 . . . 4 (𝑦 = (𝐹𝑥) → (𝜓𝜑))
54ralrn 6328 . . 3 (𝐹 Fn 𝐴 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜑))
61, 5syl 17 . 2 (𝐹:𝐴onto𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜑))
7 forn 6085 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87raleqdv 3137 . 2 (𝐹:𝐴onto𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑦𝐵 𝜓))
96, 8bitr3d 270 1 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wral 2908  ran crn 5085   Fn wfn 5852  ontowfo 5855  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865
This theorem is referenced by:  cbvexfo  6510  cocan2  6512  f1oweALT  7112  supisolem  8339  qtopeu  21459  deg1leb  23793  dchrelbas4  24902  cnpconn  30973  cocanfo  33183
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