![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5452 | . . 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 2190 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝜓 ↔ 𝜑)) |
5 | 4 | rexrn 5666 | . . 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 5453 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
8 | 7 | rexeqdv 2690 | . 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 1363 ∃wrex 2466 ran crn 4639 Fn wfn 5223 –onto→wfo 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) |
Copyright terms: Public domain | W3C validator |