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Theorem cbvfo 7280
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 6798 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 cbvfo.1 . . . . . 6 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
32bicomd 222 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜓𝜑))
43eqcoms 2732 . . . 4 (𝑦 = (𝐹𝑥) → (𝜓𝜑))
54ralrn 7080 . . 3 (𝐹 Fn 𝐴 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜑))
61, 5syl 17 . 2 (𝐹:𝐴onto𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜑))
7 forn 6799 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87raleqdv 3317 . 2 (𝐹:𝐴onto𝐵 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑦𝐵 𝜓))
96, 8bitr3d 281 1 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wral 3053  ran crn 5668   Fn wfn 6529  ontowfo 6532  cfv 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542
This theorem is referenced by:  cbvexfo  7281  cocan2  7283  f1oweALT  7953  supisolem  9465  qtopeu  23564  deg1leb  25975  dchrelbas4  27116  cnpconn  34738  cocanfo  37090  aks6d1c1p5  41479
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