![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > funcestrcsetclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for funcestrcsetc 18205. (Contributed by AV, 22-Mar-2020.) |
Ref | Expression |
---|---|
funcestrcsetc.e | ⊢ 𝐸 = (ExtStrCat‘𝑈) |
funcestrcsetc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcestrcsetc.b | ⊢ 𝐵 = (Base‘𝐸) |
funcestrcsetc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcestrcsetc.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
funcestrcsetc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) |
Ref | Expression |
---|---|
funcestrcsetclem2 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcestrcsetc.e | . . 3 ⊢ 𝐸 = (ExtStrCat‘𝑈) | |
2 | funcestrcsetc.s | . . 3 ⊢ 𝑆 = (SetCat‘𝑈) | |
3 | funcestrcsetc.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
4 | funcestrcsetc.c | . . 3 ⊢ 𝐶 = (Base‘𝑆) | |
5 | funcestrcsetc.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | funcestrcsetc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) | |
7 | 1, 2, 3, 4, 5, 6 | funcestrcsetclem1 18196 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) |
8 | 1, 3, 5 | estrcbasbas 18186 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (Base‘𝑋) ∈ 𝑈) |
9 | 7, 8 | eqeltrd 2839 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 WUnicwun 10738 Basecbs 17245 SetCatcsetc 18129 ExtStrCatcestrc 18177 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-wun 10740 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-hom 17322 df-cco 17323 df-estrc 18178 |
This theorem is referenced by: funcestrcsetclem8 18203 funcestrcsetclem9 18204 fullestrcsetc 18207 |
Copyright terms: Public domain | W3C validator |