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Mirrors > Home > MPE Home > Th. List > fvmpti | Structured version Visualization version GIF version |
Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvmptg.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmpti | ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | fvmptg.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
3 | 1, 2 | fvmptg 6531 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = 𝐶) |
4 | fvi 6506 | . . . 4 ⊢ (𝐶 ∈ V → ( I ‘𝐶) = 𝐶) | |
5 | 4 | adantl 475 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → ( I ‘𝐶) = 𝐶) |
6 | 3, 5 | eqtr4d 2864 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
7 | 1 | eleq1d 2891 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
8 | 2 | dmmpt 5875 | . . . . . . . 8 ⊢ dom 𝐹 = {𝑥 ∈ 𝐷 ∣ 𝐵 ∈ V} |
9 | 7, 8 | elrab2 3589 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝐹 ↔ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ V)) |
10 | 9 | baib 531 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ dom 𝐹 ↔ 𝐶 ∈ V)) |
11 | 10 | notbid 310 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐶 ∈ V)) |
12 | ndmfv 6467 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
13 | 11, 12 | syl6bir 246 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)) |
14 | 13 | imp 397 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ∅) |
15 | fvprc 6430 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ( I ‘𝐶) = ∅) | |
16 | 15 | adantl 475 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → ( I ‘𝐶) = ∅) |
17 | 14, 16 | eqtr4d 2864 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ V) → (𝐹‘𝐴) = ( I ‘𝐶)) |
18 | 6, 17 | pm2.61dan 847 | 1 ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4146 ↦ cmpt 4954 I cid 5251 dom cdm 5346 ‘cfv 6127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fv 6135 |
This theorem is referenced by: fvmpt2i 6542 fvmptex 6546 sumeq2ii 14807 summolem3 14829 fsumf1o 14838 isumshft 14952 prodeq2ii 15023 prodmolem3 15043 fprodf1o 15056 |
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