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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 6884 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 6884 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵)) | |
2 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
3 | ffnfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | ffnfvf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
6 | 4, 5 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
7 | ffnfvf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2994 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵 |
9 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵 | |
10 | fveq2 6672 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
11 | 10 | eleq1d 2899 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 2, 3, 8, 9, 11 | cbvralfw 3439 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
13 | 12 | anbi2i 624 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
14 | 1, 13 | bitri 277 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 |
This theorem is referenced by: ixpf 8486 fconst7 41546 |
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