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Theorem rdgss 6384
Description: Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
Hypotheses
Ref Expression
rdgss.1 (𝜑𝐹 Fn V)
rdgss.2 (𝜑𝐼𝑉)
rdgss.3 (𝜑𝐴 ∈ On)
rdgss.4 (𝜑𝐵 ∈ On)
rdgss.5 (𝜑𝐴𝐵)
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.
StepHypRef Expression
1 rdgss.5 . . . 4 (𝜑𝐴𝐵)
2 ssel 3150 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssid 3176 . . . . . . 7 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4 fveq2 5516 . . . . . . . . . 10 (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
54fveq2d 5520 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
65sseq2d 3186 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
76rspcev 2842 . . . . . . 7 ((𝑥𝐵 ∧ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) → ∃𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
83, 7mpan2 425 . . . . . 6 (𝑥𝐵 → ∃𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
92, 8syl6 33 . . . . 5 (𝐴𝐵 → (𝑥𝐴 → ∃𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
109ralrimiv 2549 . . . 4 (𝐴𝐵 → ∀𝑥𝐴𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
111, 10syl 14 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
12 iunss2 3932 . . 3 (∀𝑥𝐴𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) → 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
13 unss2 3307 . . 3 ( 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) → (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) ⊆ (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
1411, 12, 133syl 17 . 2 (𝜑 → (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))) ⊆ (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
15 rdgss.1 . . 3 (𝜑𝐹 Fn V)
16 rdgss.2 . . 3 (𝜑𝐼𝑉)
17 rdgss.3 . . 3 (𝜑𝐴 ∈ On)
18 rdgival 6383 . . 3 ((𝐹 Fn V ∧ 𝐼𝑉𝐴 ∈ On) → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
1915, 16, 17, 18syl3anc 1238 . 2 (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20 rdgss.4 . . 3 (𝜑𝐵 ∈ On)
21 rdgival 6383 . . 3 ((𝐹 Fn V ∧ 𝐼𝑉𝐵 ∈ On) → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
2215, 16, 20, 21syl3anc 1238 . 2 (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
2314, 19, 223sstr4d 3201 1 (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2738  cun 3128  wss 3130   ciun 3887  Oncon0 4364   Fn wfn 5212  cfv 5217  reccrdg 6370
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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306  df-irdg 6371
This theorem is referenced by:  oawordi  6470
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