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Mirrors > Home > MPE Home > Th. List > tfrlem6 | Structured version Visualization version GIF version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem6 | ⊢ Rel recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 5693 | . . 3 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑔 ∈ 𝐴 Rel 𝑔) | |
2 | tfrlem.1 | . . . . 5 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
3 | 2 | tfrlem4 8017 | . . . 4 ⊢ (𝑔 ∈ 𝐴 → Fun 𝑔) |
4 | funrel 6374 | . . . 4 ⊢ (Fun 𝑔 → Rel 𝑔) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑔 ∈ 𝐴 → Rel 𝑔) |
6 | 1, 5 | mprgbir 3155 | . 2 ⊢ Rel ∪ 𝐴 |
7 | 2 | recsfval 8019 | . . 3 ⊢ recs(𝐹) = ∪ 𝐴 |
8 | 7 | releqi 5654 | . 2 ⊢ (Rel recs(𝐹) ↔ Rel ∪ 𝐴) |
9 | 6, 8 | mpbir 233 | 1 ⊢ Rel recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 ∪ cuni 4840 ↾ cres 5559 Rel wrel 5562 Oncon0 6193 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 recscrecs 8009 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-wrecs 7949 df-recs 8010 |
This theorem is referenced by: tfrlem7 8021 tfrlem11 8026 tfrlem15 8030 tfrlem16 8031 |
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