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| Mirrors > Home > MPE Home > Th. List > Mathboxes > discsubclem | Structured version Visualization version GIF version | ||
| Description: Lemma for discsubc 49870. (Contributed by Zhi Wang, 1-Nov-2025.) |
| Ref | Expression |
|---|---|
| discsubc.j | ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) |
| Ref | Expression |
|---|---|
| discsubclem | ⊢ 𝐽 Fn (𝑆 × 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | discsubc.j | . 2 ⊢ 𝐽 = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅)) | |
| 2 | snex 5409 | . . 3 ⊢ {(𝐼‘𝑥)} ∈ V | |
| 3 | 0ex 5269 | . . 3 ⊢ ∅ ∈ V | |
| 4 | 2, 3 | ifex 4537 | . 2 ⊢ if(𝑥 = 𝑦, {(𝐼‘𝑥)}, ∅) ∈ V |
| 5 | 1, 4 | fnmpoi 8065 | 1 ⊢ 𝐽 Fn (𝑆 × 𝑆) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: = wceq 1569 ∅c0 4285 ifcif 4486 {csn 4588 × cxp 5658 Fn wfn 6531 ‘cfv 6536 ∈ cmpo 7414 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403 ax-un 7734 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 |
| This theorem is used by: discsubc 49870 iinfconstbaslem 49871 |
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