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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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