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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 1843 | . . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵 | |
2 | nfcsb1v 3549 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
3 | fmptf.1 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfel 2777 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 |
5 | csbeq1a 3542 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
6 | 5 | eleq1d 2686 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)) |
7 | 1, 4, 6 | cbvral 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵) |
8 | fmptf.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
9 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
10 | 9, 2, 5 | cbvmpt 4749 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
11 | 8, 10 | eqtri 2644 | . . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
12 | 11 | fmpt 6381 | . 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
13 | 7, 12 | bitri 264 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ⦋csb 3533 ↦ cmpt 4729 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 |
This theorem is referenced by: rnmptssf 39462 |
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