Theorem rdgss 6592

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 3222	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		ssid 3248	. . . . . . 7 ⊢ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
5	4	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
6	5	sseq2d 3258	. . . . . . . 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 2911	. . . . . . 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 2605	. . . 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 4020	. . 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 3380	. . 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 6591	. . 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 1274	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 ∪ ∪ 𝑥 ∈ 𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20		rdgss.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
21		rdgival 6591	. . 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 1274	. 2 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 ∪ ∪ 𝑦 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
23	14, 19, 22	3sstr4d 3273	1 ⊢ (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  ∪ ciun 3975  Oncon0 4466   Fn wfn 5328  ‘cfv 5333  reccrdg 6578

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514  df-irdg 6579

This theorem is referenced by:  oawordi  6680