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Mirrors > Home > MPE Home > Th. List > fvmptd4 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 7015 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.) |
Ref | Expression |
---|---|
fvmptd4.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptd4.2 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
fvmptd4.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd4.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmptd4 | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd4.2 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
2 | fvmptd4.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
3 | 2 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
4 | fvmptd4.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
5 | fvmptd4.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
6 | 1, 3, 4, 5 | fvmptd 7022 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ↦ cmpt 5230 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: selvval 22156 mhpmulcl 22170 psdval 22180 psdcoef 22181 evl1deg1 33580 evl1deg2 33581 evl1deg3 33582 2sqr3minply 33752 evlsvval 42546 evlsvvval 42549 evlsvarval 42551 selvvvval 42571 prjcrvval 42618 dvnprodlem1 45901 uspgrimprop 47810 |
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