Theorem setrecsss 44823

Description: The setrecs operator respects the subset relation between two functions 𝐹 and 𝐺. (Contributed by Emmett Weisz, 13-Mar-2022.)

Hypotheses

Ref	Expression
setrecsss.1	⊢ (𝜑 → Fun 𝐺)
setrecsss.2	⊢ (𝜑 → 𝐹 ⊆ 𝐺)

Assertion

Ref	Expression
setrecsss	⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))

Proof of Theorem setrecsss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2821	. 2 ⊢ setrecs(𝐹) = setrecs(𝐹)
2		setrecsss.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ 𝐺)
3		imass1 5964	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥}))
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ {𝑥}) ⊆ (𝐺 “ {𝑥}))
5	4	unissd 4848	. . . . . . 7 ⊢ (𝜑 → ∪ (𝐹 “ {𝑥}) ⊆ ∪ (𝐺 “ {𝑥}))
6		setrecsss.1	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
7		funss 6374	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (Fun 𝐺 → Fun 𝐹))
8	2, 6, 7	sylc 65	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
9		funfv 6750	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥}))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑥) = ∪ (𝐹 “ {𝑥}))
11		funfv 6750	. . . . . . . 8 ⊢ (Fun 𝐺 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥}))
12	6, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑥) = ∪ (𝐺 “ {𝑥}))
13	5, 10, 12	3sstr4d 4014	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑥) ⊆ (𝐺‘𝑥))
14	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ (𝐺‘𝑥))
15		eqid 2821	. . . . . 6 ⊢ setrecs(𝐺) = setrecs(𝐺)
16		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ∈ V)
18		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → 𝑥 ⊆ setrecs(𝐺))
19	15, 17, 18	setrec1 44814	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐺‘𝑥) ⊆ setrecs(𝐺))
20	14, 19	sstrd 3977	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs(𝐺)) → (𝐹‘𝑥) ⊆ setrecs(𝐺))
21	20	ex 415	. . 3 ⊢ (𝜑 → (𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺)))
22	21	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs(𝐺) → (𝐹‘𝑥) ⊆ setrecs(𝐺)))
23	1, 22	setrec2v 44819	1 ⊢ (𝜑 → setrecs(𝐹) ⊆ setrecs(𝐺))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  {csn 4567  ∪ cuni 4838   “ cima 5558  Fun wfun 6349  ‘cfv 6355  setrecscsetrecs 44806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-r1 9193  df-rank 9194  df-setrecs 44807

This theorem is referenced by:  setrecsres  44824