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Theorem rdgss 5970
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 2939 . . . . . 6 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
3 ssid 2964 . . . . . . 7 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))
4 fveq2 5178 . . . . . . . . . 10 (𝑦 = 𝑥 → (rec(𝐹, 𝐼)‘𝑦) = (rec(𝐹, 𝐼)‘𝑥))
54fveq2d 5182 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) = (𝐹‘(rec(𝐹, 𝐼)‘𝑥)))
65sseq2d 2973 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)) ↔ (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
76rspcev 2656 . . . . . . 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 401 . . . . . 6 (𝑥𝐵 → ∃𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦)))
92, 8syl6 29 . . . . 5 (𝐴𝐵 → (𝑥𝐴 → ∃𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑥)) ⊆ (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
109ralrimiv 2391 . . . 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 3702 . . 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 3114 . . 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 5969 . . 3 ((𝐹 Fn V ∧ 𝐼𝑉𝐴 ∈ On) → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
1915, 16, 17, 18syl3anc 1135 . 2 (𝜑 → (rec(𝐹, 𝐼)‘𝐴) = (𝐼 𝑥𝐴 (𝐹‘(rec(𝐹, 𝐼)‘𝑥))))
20 rdgss.4 . . 3 (𝜑𝐵 ∈ On)
21 rdgival 5969 . . 3 ((𝐹 Fn V ∧ 𝐼𝑉𝐵 ∈ On) → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
2215, 16, 20, 21syl3anc 1135 . 2 (𝜑 → (rec(𝐹, 𝐼)‘𝐵) = (𝐼 𝑦𝐵 (𝐹‘(rec(𝐹, 𝐼)‘𝑦))))
2314, 19, 223sstr4d 2988 1 (𝜑 → (rec(𝐹, 𝐼)‘𝐴) ⊆ (rec(𝐹, 𝐼)‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  wral 2306  wrex 2307  Vcvv 2557  cun 2915  wss 2917   ciun 3657  Oncon0 4100   Fn wfn 4897  cfv 4902  reccrdg 5956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957
This theorem is referenced by:  oawordi  6049
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