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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 2737 | . 2 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
| 2 | setrecsss.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ 𝐺) | |
| 3 | imass1 6119 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) | |
| 4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥})) |
| 5 | 4 | unissd 4917 | . . . . . . 7 ⊢ (𝜑 → ∪ (𝐹 “ {𝑥}) ⊆ ∪ (𝐺 “ {𝑥})) |
| 6 | setrecsss.1 | . . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺) | |
| 7 | funss 6585 | . . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹)) | |
| 8 | 2, 6, 7 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → Fun 𝐹) |
| 9 | funfv 6996 | . . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) | |
| 10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥})) |
| 11 | funfv 6996 | . . . . . . . 8 ⊢ (Fun 𝐺 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) | |
| 12 | 6, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥})) |
| 13 | 5, 10, 12 | 3sstr4d 4039 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
| 14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ (𝐺‘𝑥)) |
| 15 | eqid 2737 | . . . . . 6 ⊢ setrecs(𝐺) = setrecs(𝐺) | |
| 16 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ∈ V) |
| 18 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ⊆ setrecs(𝐺)) | |
| 19 | 15, 17, 18 | setrec1 49210 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐺‘𝑥) ⊆ setrecs(𝐺)) |
| 20 | 14, 19 | sstrd 3994 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ setrecs(𝐺)) |
| 21 | 20 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
| 22 | 21 | alrimiv 1927 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺))) |
| 23 | 1, 22 | setrec2v 49215 | 1 ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 “ cima 5688 Fun wfun 6555 ‘cfv 6561 setrecscsetrecs 49202 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 df-setrecs 49203 |
| This theorem is referenced by: setrecsres 49221 |
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