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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmptf | Structured version Visualization version GIF version |
Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fmptf.1 | ⊢ Ⅎ𝑥𝐵 |
fmptf.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Ref | Expression |
---|---|
fmptf | ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵 | |
2 | nfcsb1v 3852 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
3 | fmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfel 2969 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 |
5 | csbeq1a 3842 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
6 | 5 | eleq1d 2874 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)) |
7 | 1, 4, 6 | cbvralw 3387 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵) |
8 | fmptf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
10 | 9, 2, 5 | cbvmpt 5131 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
11 | 8, 10 | eqtri 2821 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
12 | 11 | fmpt 6851 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
13 | 7, 12 | bitri 278 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 ⦋csb 3828 ↦ cmpt 5110 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 |
This theorem is referenced by: rnmptssf 41885 |
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