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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 7128 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 7128 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵)) | |
2 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑧𝐴 | |
3 | ffnfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | ffnfvf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2891 | . . . . . 6 ⊢ Ⅎ𝑥𝑧 | |
6 | 4, 5 | nffv 6906 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
7 | ffnfvf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfel 2906 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵 |
9 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵 | |
10 | fveq2 6896 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
11 | 10 | eleq1d 2810 | . . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
12 | 2, 3, 8, 9, 11 | cbvralfw 3291 | . . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
13 | 12 | anbi2i 621 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
14 | 1, 13 | bitri 274 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2098 Ⅎwnfc 2875 ∀wral 3050 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 |
This theorem is referenced by: ixpf 8939 fconst7 44781 |
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