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Mirrors > Home > ILE Home > Th. List > rexrn | GIF version |
Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
Ref | Expression |
---|---|
rexrn.1 | ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexrn | ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvex 5212 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ V) | |
2 | 1 | funfni 5019 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V) |
3 | fvelrnb 5242 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥)) | |
4 | eqcom 2083 | . . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
5 | 4 | rexbii 2373 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) |
6 | 3, 5 | syl6bb 194 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))) |
7 | rexrn.1 | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
8 | 7 | adantl 271 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) |
9 | 2, 6, 8 | rexxfr2d 4215 | 1 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 Vcvv 2601 ran crn 4364 Fn wfn 4917 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: elrnrexdm 5327 rexrnmpt 5331 cbvexfo 5446 rexanuz 9874 |
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