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Mirrors > Home > ILE Home > Th. List > ffnfvf | GIF version |
Description: A function maps to a class to which all values belong. This version of ffnfv 5546 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
ffnfvf.1 | ⊢ Ⅎ𝑥𝐴 |
ffnfvf.2 | ⊢ Ⅎ𝑥𝐵 |
ffnfvf.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
ffnfvf | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffnfv 5546 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵)) | |
2 | nfcv 2258 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
3 | ffnfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | ffnfvf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2258 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
6 | 4, 5 | nffv 5399 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
7 | ffnfvf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2267 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵 |
9 | nfv 1493 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵 | |
10 | fveq2 5389 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
11 | 10 | eleq1d 2186 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 2, 3, 8, 9, 11 | cbvralf 2625 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
13 | 12 | anbi2i 452 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
14 | 1, 13 | bitri 183 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1465 Ⅎwnfc 2245 ∀wral 2393 Fn wfn 5088 ⟶wf 5089 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: ixpf 6582 |
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