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Theorem cbvexfo 5687
 Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
Hypothesis
Ref Expression
cbvfo.1 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvexfo (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 cbvexfo
StepHypRef Expression
1 fofn 5347 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 cbvfo.1 . . . . . 6 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
32bicomd 140 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜓𝜑))
43eqcoms 2142 . . . 4 (𝑦 = (𝐹𝑥) → (𝜓𝜑))
54rexrn 5557 . . 3 (𝐹 Fn 𝐴 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
61, 5syl 14 . 2 (𝐹:𝐴onto𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
7 forn 5348 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87rexeqdv 2633 . 2 (𝐹:𝐴onto𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑦𝐵 𝜓))
96, 8bitr3d 189 1 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wrex 2417  ran crn 4540   Fn wfn 5118  –onto→wfo 5121  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131 This theorem is referenced by: (None)
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