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Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrecsss | Structured version Visualization version GIF version |
Description: The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.) |
Ref | Expression |
---|---|
setrecsss.1 | ⊢ (𝜑 → Fun 𝐺) |
setrecsss.2 | ⊢ (𝜑 → 𝐹 ⊆ 𝐺) |
Ref | Expression |
---|---|
setrecsss | ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . 2 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | setrecsss.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
3 | imass1 6100 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) |
5 | 4 | unissd 4913 | . . . . . . 7 ⊢ (𝜑 → ∪ (𝐹 “ {𝑥}) ⊆ ∪ (𝐺 “ {𝑥})) |
6 | setrecsss.1 | . . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺) | |
7 | funss 6567 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
8 | 2, 6, 7 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
9 | funfv 6980 | . . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) |
11 | funfv 6980 | . . . . . . . 8 ⊢ (Fun 𝐺 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) | |
12 | 6, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) |
13 | 5, 10, 12 | 3sstr4d 4020 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
14 | 13 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
15 | eqid 2725 | . . . . . 6 ⊢ setrecs(𝐺) = setrecs(𝐺) | |
16 | vex 3467 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ∈ V) |
18 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ⊆ setrecs(𝐺)) | |
19 | 15, 17, 18 | setrec1 48234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐺‘𝑥) ⊆ setrecs(𝐺)) |
20 | 14, 19 | sstrd 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ setrecs(𝐺)) |
21 | 20 | ex 411 | . . 3 ⊢ (𝜑 → (𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
22 | 21 | alrimiv 1922 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
23 | 1, 22 | setrec2v 48239 | 1 ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3939 {csn 4624 ∪ cuni 4903 “ cima 5675 Fun wfun 6537 ‘cfv 6543 setrecscsetrecs 48226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-reg 9615 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-r1 9787 df-rank 9788 df-setrecs 48227 |
This theorem is referenced by: setrecsres 48245 |
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