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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptd4 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 6775 (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 485 | . 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 6782 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5110 ‘cfv 6339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 |
This theorem is referenced by: evlsvarval 39853 evlsbagval 39854 |
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