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Theorem cbvexfo 5803
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 5455 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 cbvfo.1 . . . . . 6 ((𝐹𝑥) = 𝑦 → (𝜑𝜓))
32bicomd 141 . . . . 5 ((𝐹𝑥) = 𝑦 → (𝜓𝜑))
43eqcoms 2192 . . . 4 (𝑦 = (𝐹𝑥) → (𝜓𝜑))
54rexrn 5669 . . 3 (𝐹 Fn 𝐴 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
61, 5syl 14 . 2 (𝐹:𝐴onto𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜑))
7 forn 5456 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87rexeqdv 2693 . 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 1364  wrex 2469  ran crn 4642   Fn wfn 5226  ontowfo 5229  cfv 5231
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239
This theorem is referenced by: (None)
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