Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > funcsetcestrclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for funcsetcestrc 16804. (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 481 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐹 = (𝑥 ∈ 𝐶 ↦ {〈(Base‘ndx), 𝑥〉})) |
3 | opeq2 4403 | . . . 4 ⊢ (𝑥 = 𝑋 → 〈(Base‘ndx), 𝑥〉 = 〈(Base‘ndx), 𝑋〉) | |
4 | 3 | sneqd 4189 | . . 3 ⊢ (𝑥 = 𝑋 → {〈(Base‘ndx), 𝑥〉} = {〈(Base‘ndx), 𝑋〉}) |
5 | 4 | adantl 482 | . 2 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑥 = 𝑋) → {〈(Base‘ndx), 𝑥〉} = {〈(Base‘ndx), 𝑋〉}) |
6 | simpr 477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
7 | snex 4908 | . . 3 ⊢ {〈(Base‘ndx), 𝑋〉} ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {〈(Base‘ndx), 𝑋〉} ∈ V) |
9 | 2, 5, 6, 8 | fvmptd 6288 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐹‘𝑋) = {〈(Base‘ndx), 𝑋〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 ↦ cmpt 4729 ‘cfv 5888 ndxcnx 15854 Basecbs 15857 SetCatcsetc 16725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: funcsetcestrclem2 16795 embedsetcestrclem 16797 funcsetcestrclem7 16801 funcsetcestrclem8 16802 funcsetcestrclem9 16803 fullsetcestrc 16806 |
Copyright terms: Public domain | W3C validator |