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| Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.) | 
| Ref | Expression | 
|---|---|
| rdglem1 | ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
| 2 | 1 | tfrlem3 8418 | . 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)))} | 
| 3 | fveq2 6906 | . . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑔‘𝑣) = (𝑔‘𝑤)) | |
| 4 | reseq2 5992 | . . . . . . . 8 ⊢ (𝑣 = 𝑤 → (𝑔 ↾ 𝑣) = (𝑔 ↾ 𝑤)) | |
| 5 | 4 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝐺‘(𝑔 ↾ 𝑣)) = (𝐺‘(𝑔 ↾ 𝑤))) | 
| 6 | 3, 5 | eqeq12d 2753 | . . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) | 
| 7 | 6 | cbvralvw 3237 | . . . . 5 ⊢ (∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) | 
| 8 | 7 | anbi2i 623 | . . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) | 
| 9 | 8 | rexbii 3094 | . . 3 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣))) ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) | 
| 10 | 9 | abbii 2809 | . 2 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑣 ∈ 𝑧 (𝑔‘𝑣) = (𝐺‘(𝑔 ↾ 𝑣)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))} | 
| 11 | 2, 10 | eqtri 2765 | 1 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 {cab 2714 ∀wral 3061 ∃wrex 3070 ↾ cres 5687 Oncon0 6384 Fn wfn 6556 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 | 
| This theorem is referenced by: rdgseg 8462 | 
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