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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 2733 | . 2 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | setrecsss.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
3 | imass1 6054 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) |
5 | 4 | unissd 4876 | . . . . . . 7 ⊢ (𝜑 → ∪ (𝐹 “ {𝑥}) ⊆ ∪ (𝐺 “ {𝑥})) |
6 | setrecsss.1 | . . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺) | |
7 | funss 6521 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
8 | 2, 6, 7 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
9 | funfv 6929 | . . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) |
11 | funfv 6929 | . . . . . . . 8 ⊢ (Fun 𝐺 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) | |
12 | 6, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) |
13 | 5, 10, 12 | 3sstr4d 3992 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
14 | 13 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
15 | eqid 2733 | . . . . . 6 ⊢ setrecs(𝐺) = setrecs(𝐺) | |
16 | vex 3448 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ∈ V) |
18 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ⊆ setrecs(𝐺)) | |
19 | 15, 17, 18 | setrec1 47222 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐺‘𝑥) ⊆ setrecs(𝐺)) |
20 | 14, 19 | sstrd 3955 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ setrecs(𝐺)) |
21 | 20 | ex 414 | . . 3 ⊢ (𝜑 → (𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
22 | 21 | alrimiv 1931 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
23 | 1, 22 | setrec2v 47227 | 1 ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 {csn 4587 ∪ cuni 4866 “ cima 5637 Fun wfun 6491 ‘cfv 6497 setrecscsetrecs 47214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-reg 9533 ax-inf2 9582 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-r1 9705 df-rank 9706 df-setrecs 47215 |
This theorem is referenced by: setrecsres 47233 |
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