Theorem rdgss 6280

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 3091	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		ssid 3117	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4		fveq2 5421	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
5	4	fveq2d 5425	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
6	5	sseq2d 3127	. . . . . . . 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 2789	. . . . . . 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 421	. . . . . 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 2504	. . . 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 3858	. . 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 3247	. . 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 6279	. . 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 1216	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20		rdgss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
21		rdgival 6279	. . 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 1216	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
23	14, 19, 22	3sstr4d 3142	1 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ∪ cun 3069   ⊆ wss 3071  ∪ ciun 3813  Oncon0 4285   Fn wfn 5118  ‘cfv 5123  reccrdg 6266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202  df-irdg 6267

This theorem is referenced by:  oawordi  6365