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Mirrors > Home > MPE Home > Th. List > wfrrel | Structured version Visualization version GIF version |
Description: The well-ordered recursion generator generates a relation. Avoids the axiom of replacement. (Contributed by Scott Fenton, 8-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.) |
Ref | Expression |
---|---|
wfrrel.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfrrel | ⊢ Rel 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrrel.1 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
2 | df-wrecs 8120 | . . 3 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
3 | 1, 2 | eqtri 2768 | . 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
4 | 3 | frrrel 8114 | 1 ⊢ Rel 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∘ ccom 5594 Rel wrel 5595 2nd c2nd 7824 frecscfrecs 8088 wrecscwrecs 8119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-iota 6390 df-fun 6434 df-fn 6435 df-fv 6440 df-ov 7275 df-frecs 8089 df-wrecs 8120 |
This theorem is referenced by: (None) |
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