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Mirrors > Home > ILE Home > Th. List > tfrlem3-2d | GIF version |
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Ref | Expression |
---|---|
tfrlem3-2d.1 | ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) |
Ref | Expression |
---|---|
tfrlem3-2d | ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem3-2d.1 | . . 3 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) | |
2 | fveq2 5178 | . . . . . 6 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔)) | |
3 | 2 | eleq1d 2106 | . . . . 5 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V)) |
4 | 3 | anbi2d 437 | . . . 4 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))) |
5 | 4 | cbvalv 1794 | . . 3 ⊢ (∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
6 | 1, 5 | sylib 127 | . 2 ⊢ (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
7 | 6 | 19.21bi 1450 | 1 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∈ wcel 1393 Vcvv 2557 Fun wfun 4896 ‘cfv 4902 |
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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 |
This theorem is referenced by: tfrlemisucfn 5938 tfrlemisucaccv 5939 tfrlemibxssdm 5941 tfrlemibfn 5942 tfrlemi14d 5947 |
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