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Mirrors > Home > MPE Home > Th. List > wfrlem2 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Scott Fenton, 21-Apr-2011.) |
Ref | Expression |
---|---|
wfrlem1.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
Ref | Expression |
---|---|
wfrlem2 | ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrlem1.1 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
2 | 1 | wfrlem1 7650 | . . 3 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))} |
3 | 2 | abeq2i 2910 | . 2 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
4 | fnfun 6197 | . . . 4 ⊢ (𝑔 Fn 𝑧 → Fun 𝑔) | |
5 | 4 | 3ad2ant1 1164 | . . 3 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → Fun 𝑔) |
6 | 5 | exlimiv 2026 | . 2 ⊢ (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → Fun 𝑔) |
7 | 3, 6 | sylbi 209 | 1 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∃wex 1875 ∈ wcel 2157 {cab 2783 ∀wral 3087 ⊆ wss 3767 ↾ cres 5312 Predcpred 5895 Fun wfun 6093 Fn wfn 6094 ‘cfv 6099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-iota 6062 df-fun 6101 df-fn 6102 df-fv 6107 |
This theorem is referenced by: wfrlem4 7654 wfrlem4OLD 7655 wfrlem5 7656 wfrrel 7657 |
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