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Mirrors > Home > MPE Home > Th. List > suppimacnvss | Structured version Visualization version GIF version |
Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7893. (Contributed by AV, 7-Apr-2019.) |
Ref | Expression |
---|---|
suppimacnvss | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exsimpl 1876 | . . . . 5 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → ∃𝑦 𝑥𝑅𝑦) | |
2 | pm5.1 824 | . . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)) | |
3 | 2 | eximi 1842 | . . . . 5 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)) |
4 | 1, 3 | jca 515 | . . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))) |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍) → (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)))) |
6 | 5 | ss2abdv 3967 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} ⊆ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))}) |
7 | cnvimadfsn 7903 | . . 3 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} | |
8 | 7 | a1i 11 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}) |
9 | suppvalbr 7896 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))}) | |
10 | 6, 8, 9 | 3sstr4d 3938 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 {cab 2712 ≠ wne 2935 Vcvv 3401 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 class class class wbr 5043 ◡ccnv 5539 “ cima 5543 (class class class)co 7202 supp csupp 7892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-supp 7893 |
This theorem is referenced by: suppimacnv 7905 fsuppinisegfi 30713 |
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