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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 6705 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 6705 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵)) | |
2 | nfcv 2932 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
3 | ffnfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | ffnfvf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2932 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
6 | 4, 5 | nffv 6509 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
7 | ffnfvf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2944 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵 |
9 | nfv 1873 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵 | |
10 | fveq2 6499 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
11 | 10 | eleq1d 2850 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 2, 3, 8, 9, 11 | cbvralf 3377 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
13 | 12 | anbi2i 613 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
14 | 1, 13 | bitri 267 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∈ wcel 2050 Ⅎwnfc 2916 ∀wral 3088 Fn wfn 6183 ⟶wf 6184 ‘cfv 6188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-fv 6196 |
This theorem is referenced by: ixpf 8281 fconst7 40974 |
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