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Mirrors > Home > MPE Home > Th. List > suppcofnd | Structured version Visualization version GIF version |
Description: The support of the composition of two functions. (Contributed by SN, 15-Sep-2023.) |
Ref | Expression |
---|---|
suppcofnd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
suppcofnd.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
suppcofnd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
suppcofnd.g | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
suppcofnd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
suppcofnd | ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppcofnd.f | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | suppcofnd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 1, 2 | fnexd 6974 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | suppcofnd.g | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
5 | suppcofnd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
6 | 4, 5 | fnexd 6974 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
7 | suppco 7863 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) | |
8 | 3, 6, 7 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍))) |
9 | fncnvima2 6824 | . . 3 ⊢ (𝐺 Fn 𝐵 → (◡𝐺 “ (𝐹 supp 𝑍)) = {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)}) | |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐺 “ (𝐹 supp 𝑍)) = {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)}) |
11 | suppcofnd.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
12 | elsuppfn 7831 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → ((𝐺‘𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍))) | |
13 | 1, 2, 11, 12 | syl3anc 1366 | . . 3 ⊢ (𝜑 → ((𝐺‘𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍))) |
14 | 13 | rabbidv 3477 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ (𝐺‘𝑥) ∈ (𝐹 supp 𝑍)} = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)}) |
15 | 8, 10, 14 | 3eqtrd 2859 | 1 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) = {𝑥 ∈ 𝐵 ∣ ((𝐺‘𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑥)) ≠ 𝑍)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {crab 3141 Vcvv 3491 ◡ccnv 5547 “ cima 5551 ∘ ccom 5552 Fn wfn 6343 ‘cfv 6348 (class class class)co 7149 supp csupp 7823 |
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 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-supp 7824 |
This theorem is referenced by: mhpinvcl 20332 |
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