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Mirrors > Home > MPE Home > Th. List > Mathboxes > rdgellim | Structured version Visualization version GIF version |
Description: Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.) |
Ref | Expression |
---|---|
rdgellim | ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6656 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → (rec(𝐹, 𝐴)‘𝑦) = (rec(𝐹, 𝐴)‘𝐶)) | |
2 | 1 | eleq2d 2898 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦) ↔ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶))) |
3 | 2 | rspcev 3615 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶)) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)) |
4 | 3 | ex 415 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦))) |
5 | eliun 4909 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦) ↔ ∃𝑦 ∈ 𝐵 𝑋 ∈ (rec(𝐹, 𝐴)‘𝑦)) | |
6 | 4, 5 | syl6ibr 254 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))) |
7 | 6 | adantl 484 | . 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))) |
8 | rdglim2a 8055 | . . . 4 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦)) | |
9 | 8 | eleq2d 2898 | . . 3 ⊢ ((𝐵 ∈ On ∧ Lim 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))) |
10 | 9 | adantr 483 | . 2 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵) ↔ 𝑋 ∈ ∪ 𝑦 ∈ 𝐵 (rec(𝐹, 𝐴)‘𝑦))) |
11 | 7, 10 | sylibrd 261 | 1 ⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∪ ciun 4905 Oncon0 6177 Lim wlim 6178 ‘cfv 6341 reccrdg 8031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-wrecs 7933 df-recs 7994 df-rdg 8032 |
This theorem is referenced by: rdglimss 34674 exrecfnlem 34676 |
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