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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmpt2df | Structured version Visualization version GIF version | ||
| Description: Deduction version of fvmpt2 7026. (Contributed by Glauco Siliprandi, 24-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| fvmpt2df.1 | ⊢ Ⅎ𝑥𝐴 | 
| fvmpt2df.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | 
| fvmpt2df.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | 
| Ref | Expression | 
|---|---|
| fvmpt2df | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvmpt2df.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | 1 | fveq1i 6906 | . 2 ⊢ (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) | 
| 3 | id 22 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) | |
| 4 | fvmpt2df.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
| 5 | fvmpt2df.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 6 | 5 | fvmpt2f 7016 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) | 
| 7 | 3, 4, 6 | syl2an2 686 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) | 
| 8 | 2, 7 | eqtrid 2788 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Ⅎwnfc 2889 ↦ cmpt 5224 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: fsupdm 46862 | 
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