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Mirrors > Home > ILE Home > Th. List > cbvexfo | GIF version |
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
Ref | Expression |
---|---|
cbvfo.1 | ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvexfo | ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5455 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) | |
2 | cbvfo.1 | . . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | bicomd 141 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝜓 ↔ 𝜑)) |
4 | 3 | eqcoms 2192 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑)) |
5 | 4 | rexrn 5669 | . . 3 ⊢ (𝐹 Fn 𝐴 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
6 | 1, 5 | syl 14 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
7 | forn 5456 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | rexeqdv 2693 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜓)) |
9 | 6, 8 | bitr3d 190 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∃wrex 2469 ran crn 4642 Fn wfn 5226 –onto→wfo 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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