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Mirrors > Home > MPE Home > Th. List > fvmptd2 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 6762 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fvmptd2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
fvmptd2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
fvmptd2.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd2.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmptd2 | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
3 | fvmptd2.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
4 | fvmptd2.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
5 | fvmptd2.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
6 | 2, 3, 4, 5 | fvmptd 6769 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5138 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: fvmptopab 7203 updjudhcoinlf 9355 updjudhcoinrg 9356 lcmf0val 15960 fvprmselelfz 16374 fvprmselgcd1 16375 setcval 17331 catcval 17350 estrcval 17368 hofval 17496 yonval 17505 frmdval 18010 smndex1igid 18063 smndex1n0mnd 18071 gexval 18697 pmatcollpw3fi1lem1 21388 chfacfscmul0 21460 chfacfscmulgsum 21462 chfacfpmmul0 21464 chfacfpmmulgsum 21466 lmfval 21834 kgenval 22137 ptval 22172 utopval 22835 ustuqtoplem 22842 utopsnneiplem 22850 tusval 22869 blfvalps 22987 tmsval 23085 metuval 23153 caufval 23872 dchrval 25804 gausslemma2dlem2 25937 gausslemma2dlem3 25938 israg 26477 perpln1 26490 perpln2 26491 isperp 26492 vtxdgfval 27243 crctcsh 27596 clwlkclwwlklem2fv1 27767 clwlkclwwlklem2fv2 27768 cofmpt2 30373 carsgval 31556 bj-finsumval0 34561 cdleme31fv2 37523 iunrelexpmin1 40046 iunrelexpmin2 40050 rfovcnvf1od 40343 limsup10exlem 42046 prproropf1olem3 43661 prprval 43670 clintopval 44105 rngcval 44227 ringcval 44273 |
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