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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqfunressuc | Structured version Visualization version GIF version |
Description: Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.) |
Ref | Expression |
---|---|
eqfunressuc | ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfunresadj 32999 | . 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ (𝑋 ∪ {𝑋})) = (𝐺 ↾ (𝑋 ∪ {𝑋}))) | |
2 | df-suc 6191 | . . 3 ⊢ suc 𝑋 = (𝑋 ∪ {𝑋}) | |
3 | 2 | reseq2i 5844 | . 2 ⊢ (𝐹 ↾ suc 𝑋) = (𝐹 ↾ (𝑋 ∪ {𝑋})) |
4 | 2 | reseq2i 5844 | . 2 ⊢ (𝐺 ↾ suc 𝑋) = (𝐺 ↾ (𝑋 ∪ {𝑋})) |
5 | 1, 3, 4 | 3eqtr4g 2881 | 1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 {csn 4560 dom cdm 5549 ↾ cres 5551 suc csuc 6187 Fun wfun 6343 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 |
This theorem is referenced by: nosupbnd1lem5 33207 |
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