Theorem rdgss 6423

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 3169	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		ssid 3195	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4		fveq2 5542	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
5	4	fveq2d 5546	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
6	5	sseq2d 3205	. . . . . . . 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 2860	. . . . . . 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 2562	. . . 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 3953	. . 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 3326	. . 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 6422	. . 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 1249	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20		rdgss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
21		rdgival 6422	. . 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 1249	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
23	14, 19, 22	3sstr4d 3220	1 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ∃wrex 2469  Vcvv 2756   ∪ cun 3147   ⊆ wss 3149  ∪ ciun 3908  Oncon0 4388   Fn wfn 5237  ‘cfv 5242  reccrdg 6409

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-iord 4391  df-on 4393  df-suc 4396  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-recs 6345  df-irdg 6410

This theorem is referenced by:  oawordi  6509