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Mirrors > Home > MPE Home > Th. List > fvmptd2 | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt 6948 (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 6955 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5188 ‘cfv 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 |
This theorem is referenced by: fvmptopabOLD 7411 updjudhcoinlf 9867 updjudhcoinrg 9868 lcmf0val 16497 fvprmselelfz 16915 fvprmselgcd1 16916 setcval 17962 catcval 17985 estrcval 18010 hofval 18140 yonval 18149 frmdval 18660 smndex1igid 18713 smndex1n0mnd 18721 gexval 19358 pmatcollpw3fi1lem1 22133 chfacfscmul0 22205 chfacfscmulgsum 22207 chfacfpmmul0 22209 chfacfpmmulgsum 22211 lmfval 22581 kgenval 22884 ptval 22919 utopval 23582 ustuqtoplem 23589 utopsnneiplem 23597 tusval 23615 blfvalps 23734 tmsval 23834 metuval 23903 caufval 24637 dchrval 26580 gausslemma2dlem2 26713 gausslemma2dlem3 26714 israg 27586 perpln1 27599 perpln2 27600 isperp 27601 vtxdgfval 28362 crctcsh 28716 clwlkclwwlklem2fv1 28886 clwlkclwwlklem2fv2 28887 cofmpt2 31495 pwrssmgc 31804 frobrhm 32012 qusima 32131 elrspunidl 32143 metidval 32411 pstmval 32416 carsgval 32843 bj-rdg0gALT 35532 bj-finsumval0 35746 cdleme31fv2 38846 iunrelexpmin1 41961 iunrelexpmin2 41965 rfovcnvf1od 42257 limsup10exlem 43984 prproropf1olem3 45668 prprval 45677 clintopval 46109 rngcval 46231 ringcval 46277 1arymaptfo 46700 2arymptfv 46707 2arymaptfo 46711 ackval42 46753 |
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