Theorem rdgss 6548

Description: Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)

Hypotheses

Ref	Expression
rdgss.1	⊢ (𝜑 → 𝐹 Fn V)
rdgss.2	⊢ (𝜑 → 𝐼 ∈ 𝑉)
rdgss.3	⊢ (𝜑 → 𝐴 ∈ On)
rdgss.4	⊢ (𝜑 → 𝐵 ∈ On)
rdgss.5	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
rdgss	⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Proof of Theorem rdgss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgss.5	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssel 3221	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		ssid 3247	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
5	4	fveq2d 5643	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
6	5	sseq2d 3257	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
7	6	rspcev 2910	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) → ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
8	3, 7	mpan2 425	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
9	2, 8	syl6 33	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
10	9	ralrimiv 2604	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
11	1, 10	syl 14	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
12		iunss2 4015	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) → ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
13		unss2 3378	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) → (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) ⊆ (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
14	11, 12, 13	3syl 17	. 2 ⊢ (𝜑 → (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) ⊆ (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
15		rdgss.1	. . 3 ⊢ (𝜑 → 𝐹 Fn V)
16		rdgss.2	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
17		rdgss.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
18		rdgival 6547	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ On) → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
19	15, 16, 17, 18	syl3anc 1273	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20		rdgss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
21		rdgival 6547	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐼 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
22	15, 16, 20, 21	syl3anc 1273	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
23	14, 19, 22	3sstr4d 3272	1 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  ∪ ciun 3970  Oncon0 4460   Fn wfn 5321  ‘cfv 5326  reccrdg 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-recs 6470  df-irdg 6535

This theorem is referenced by:  oawordi  6636