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Mirrors > Home > MPE Home > Th. List > frnsuppeq | Structured version Visualization version GIF version |
Description: Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.) |
Ref | Expression |
---|---|
frnsuppeq | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fex 6992 | . . . . . . 7 ⊢ ((𝐹:𝐼⟶𝑆 ∧ 𝐼 ∈ 𝑉) → 𝐹 ∈ V) | |
2 | 1 | expcom 416 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V)) |
3 | 2 | adantr 483 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → 𝐹 ∈ V)) |
4 | 3 | imp 409 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝐹 ∈ V) |
5 | simplr 767 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → 𝑍 ∈ 𝑊) | |
6 | suppimacnv 7844 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) | |
7 | 4, 5, 6 | syl2anc 586 | . . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (V ∖ {𝑍}))) |
8 | invdif 4248 | . . . . . 6 ⊢ (𝑆 ∩ (V ∖ {𝑍})) = (𝑆 ∖ {𝑍}) | |
9 | 8 | imaeq2i 5930 | . . . . 5 ⊢ (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (𝑆 ∖ {𝑍})) |
10 | ffun 6520 | . . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → Fun 𝐹) | |
11 | inpreima 6837 | . . . . . . 7 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍})))) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍})))) |
13 | cnvimass 5952 | . . . . . . . 8 ⊢ (◡𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹 | |
14 | fdm 6525 | . . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = 𝐼) | |
15 | fimacnv 6842 | . . . . . . . . 9 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ 𝑆) = 𝐼) | |
16 | 14, 15 | eqtr4d 2862 | . . . . . . . 8 ⊢ (𝐹:𝐼⟶𝑆 → dom 𝐹 = (◡𝐹 “ 𝑆)) |
17 | 13, 16 | sseqtrid 4022 | . . . . . . 7 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆)) |
18 | sseqin2 4195 | . . . . . . 7 ⊢ ((◡𝐹 “ (V ∖ {𝑍})) ⊆ (◡𝐹 “ 𝑆) ↔ ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍}))) | |
19 | 17, 18 | sylib 220 | . . . . . 6 ⊢ (𝐹:𝐼⟶𝑆 → ((◡𝐹 “ 𝑆) ∩ (◡𝐹 “ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍}))) |
20 | 12, 19 | eqtrd 2859 | . . . . 5 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (𝑆 ∩ (V ∖ {𝑍}))) = (◡𝐹 “ (V ∖ {𝑍}))) |
21 | 9, 20 | syl5reqr 2874 | . . . 4 ⊢ (𝐹:𝐼⟶𝑆 → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍}))) |
22 | 21 | adantl 484 | . . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (◡𝐹 “ (V ∖ {𝑍})) = (◡𝐹 “ (𝑆 ∖ {𝑍}))) |
23 | 7, 22 | eqtrd 2859 | . 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝐹:𝐼⟶𝑆) → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))) |
24 | 23 | ex 415 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∖ cdif 3936 ∩ cin 3938 ⊆ wss 3939 {csn 4570 ◡ccnv 5557 dom cdm 5558 “ cima 5561 Fun wfun 6352 ⟶wf 6354 (class class class)co 7159 supp csupp 7833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-supp 7834 |
This theorem is referenced by: frnfsuppbi 8865 frnnn0supp 11956 ffs2 30467 eulerpartlemmf 31637 pwfi2f1o 39702 |
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