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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmptff | Structured version Visualization version GIF version |
Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 5-Jan-2025.) |
Ref | Expression |
---|---|
fmptff.1 | ⊢ Ⅎ𝑥𝐴 |
fmptff.2 | ⊢ Ⅎ𝑥𝐵 |
fmptff.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Ref | Expression |
---|---|
fmptff | ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptff.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2899 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1910 | . . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵 | |
4 | nfcsb1v 3917 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
5 | fmptff.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfel 2914 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 |
7 | csbeq1a 3906 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
8 | 7 | eleq1d 2814 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)) |
9 | 1, 2, 3, 6, 8 | cbvralfw 3298 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵) |
10 | fmptff.3 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
12 | 1, 2, 11, 4, 7 | cbvmptf 5257 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
13 | 10, 12 | eqtri 2756 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
14 | 13 | fmpt 7120 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
15 | 9, 14 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2879 ∀wral 3058 ⦋csb 3892 ↦ cmpt 5231 ⟶wf 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6550 df-fn 6551 df-f 6552 |
This theorem is referenced by: fvmptelcdmf 44647 fmptdff 44648 rnmptssff 44651 |
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