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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptd3 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 6282. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fvmptd3.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
fvmptd3.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptd3.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd3.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmptd3 | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd3.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
2 | fvmptd3.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
3 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
5 | fvmptd3.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
6 | fvmptd3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
7 | 3, 4, 5, 6 | fvmptf 6301 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
8 | 1, 2, 7 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 |
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-8 1992 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-pow 4843 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-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-fv 5896 |
This theorem is referenced by: limsuplt2 39985 limsupge 39993 smflimsuplem1 41026 smflimsuplem5 41030 smflimsuplem7 41032 |
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