Theorem rdgss 6386

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 3151	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		ssid 3177	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
5	4	fveq2d 5521	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
6	5	sseq2d 3187	. . . . . . . 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 2843	. . . . . . 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 2549	. . . 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 3933	. . 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 3308	. . 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 6385	. . 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 1238	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20		rdgss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
21		rdgival 6385	. . 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 1238	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
23	14, 19, 22	3sstr4d 3202	1 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2739   ∪ cun 3129   ⊆ wss 3131  ∪ ciun 3888  Oncon0 4365   Fn wfn 5213  ‘cfv 5218  reccrdg 6372

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308  df-irdg 6373

This theorem is referenced by:  oawordi  6472