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Mirrors > Home > MPE Home > Th. List > funcsetcestrclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for funcsetcestrc 17416. (Contributed by AV, 27-Mar-2020.) |
Ref | Expression |
---|---|
funcsetcestrc.s | ⊢ 𝑆 = (SetCat‘𝑈) |
funcsetcestrc.c | ⊢ 𝐶 = (Base‘𝑆) |
funcsetcestrc.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
Ref | Expression |
---|---|
funcsetcestrclem1 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcsetcestrc.f | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
3 | opeq2 4806 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈(Base‘ndx), 𝑥〉 = 〈(Base‘ndx), 𝑋〉) | |
4 | 3 | sneqd 4581 | . . 3 ⊢ (𝑥 = 𝑋 → {〈(Base‘ndx), 𝑥〉} = {〈(Base‘ndx), 𝑋〉}) |
5 | 4 | adantl 484 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑥 = 𝑋) → {〈(Base‘ndx), 𝑥〉} = {〈(Base‘ndx), 𝑋〉}) |
6 | simpr 487 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
7 | snex 5334 | . . 3 ⊢ {〈(Base‘ndx), 𝑋〉} ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ V) |
9 | 2, 5, 6, 8 | fvmptd 6777 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 ↦ cmpt 5148 ‘cfv 6357 ndxcnx 16482 Basecbs 16485 SetCatcsetc 17337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: funcsetcestrclem2 17407 embedsetcestrclem 17409 funcsetcestrclem7 17413 funcsetcestrclem8 17414 funcsetcestrclem9 17415 fullsetcestrc 17418 |
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