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Mirrors > Home > MPE Home > Th. List > mpoeq123dv | Structured version Visualization version GIF version |
Description: An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.) |
Ref | Expression |
---|---|
mpoeq123dv.1 | ⊢ (𝜑 → 𝐴 = 𝐷) |
mpoeq123dv.2 | ⊢ (𝜑 → 𝐵 = 𝐸) |
mpoeq123dv.3 | ⊢ (𝜑 → 𝐶 = 𝐹) |
Ref | Expression |
---|---|
mpoeq123dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoeq123dv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐷) | |
2 | mpoeq123dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐸) | |
3 | 2 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸) |
4 | mpoeq123dv.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐹) | |
5 | 4 | adantr 481 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹) |
6 | 1, 3, 5 | mpoeq123dva 7485 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∈ cmpo 7413 |
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-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: mpoeq123i 7487 mptmpoopabbrd 8069 mptmpoopabbrdOLD 8071 el2mpocsbcl 8073 bropopvvv 8078 bropfvvvv 8080 prdsval 17403 imasval 17459 imasvscaval 17486 homffval 17636 homfeq 17640 comfffval 17644 comffval 17645 comfffval2 17647 comffval2 17648 comfeq 17652 oppcval 17659 monfval 17681 sectffval 17699 invffval 17707 isofn 17724 cofuval 17834 natfval 17899 fucval 17912 fucco 17917 coafval 18016 setcval 18029 setcco 18035 catcval 18052 catcco 18057 estrcval 18077 estrcco 18083 xpcval 18131 1stfval 18145 2ndfval 18148 prfval 18153 evlfval 18172 evlf2 18173 curfval 18178 hofval 18207 hof2fval 18210 plusffval 18569 efmnd 18753 grpsubfval 18870 grpsubfvalALT 18871 grpsubpropd 18930 mulgfval 18954 mulgfvalALT 18955 mulgpropd 18998 lsmfval 19508 pj1fval 19564 efgtf 19592 prdsmgp 20136 dvrfval 20220 scaffval 20495 ipffval 21207 phssip 21217 frlmip 21339 psrval 21474 mamufval 21894 mvmulfval 22051 marrepfval 22069 marepvfval 22074 submafval 22088 submaval 22090 madufval 22146 minmar1fval 22155 mat2pmatfval 22232 cpm2mfval 22258 decpmatval0 22273 decpmatval 22274 pmatcollpw3lem 22292 xkoval 23098 xkopt 23166 xpstopnlem1 23320 submtmd 23615 blfvalps 23896 ishtpy 24495 isphtpy 24504 pcofval 24533 rrxip 24914 q1pval 25678 r1pval 25681 taylfval 25878 istrkgl 27747 midf 28065 ismidb 28067 ttgval 28164 ttgvalOLD 28165 wwlksnon 29143 wspthsnon 29144 clwwlknonmpo 29380 grpodivfval 29825 dipfval 29993 idlsrgval 32662 submatres 32855 lmatval 32862 lmatcl 32865 qqhval 33023 sxval 33257 sitmval 33417 cndprobval 33501 mclsval 34623 csbfinxpg 36361 rrnval 36787 ldualset 38087 paddfval 38760 tgrpfset 39707 tgrpset 39708 erngfset 39762 erngset 39763 erngfset-rN 39770 erngset-rN 39771 dvafset 39967 dvaset 39968 dvhfset 40043 dvhset 40044 djaffvalN 40096 djafvalN 40097 djhffval 40359 djhfval 40360 hlhilset 40897 eldiophb 41583 mendval 42013 mnringvald 43055 mnringmulrd 43068 hoidmvval 45378 ovnhoi 45404 hspval 45410 hspmbllem2 45428 hoimbl 45432 rngcvalALTV 46944 rngccoALTV 46971 funcrngcsetcALT 46982 ringcvalALTV 46990 ringccoALTV 47034 lincop 47173 lines 47501 rrxlines 47503 spheres 47516 |
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