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Mirrors > Home > MPE Home > Th. List > ffnfvf | Structured version Visualization version GIF version |
Description: A function maps to a class to which all values belong. This version of ffnfv 6913 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 6913 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵)) | |
2 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
3 | ffnfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | ffnfvf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
6 | 4, 5 | nffv 6705 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
7 | ffnfvf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2911 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵 |
9 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵 | |
10 | fveq2 6695 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
11 | 10 | eleq1d 2815 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 2, 3, 8, 9, 11 | cbvralfw 3334 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
13 | 12 | anbi2i 626 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
14 | 1, 13 | bitri 278 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 Ⅎwnfc 2877 ∀wral 3051 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: ixpf 8579 fconst7 42425 |
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