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Theorem cbvexfo 5800
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 5452 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 cbvfo.1 . . . . . 6 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
32bicomd 141 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜓𝜑))
43eqcoms 2190 . . . 4 (𝑦 = (𝐹𝑥) → (𝜓𝜑))
54rexrn 5666 . . 3 (𝐹 Fn 𝐴 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
61, 5syl 14 . 2 (𝐹:𝐴onto𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
7 forn 5453 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87rexeqdv 2690 . 2 (𝐹:𝐴onto𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑦𝐵 𝜓))
96, 8bitr3d 190 1 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1363  wrex 2466  ran crn 4639   Fn wfn 5223  ontowfo 5226  cfv 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236
This theorem is referenced by: (None)
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