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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2rnb | Structured version Visualization version GIF version |
Description: The alternate function value at a class 𝐴 is defined, i.e. in the range of the function, iff the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2rnb | ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funressndmafv2rn 43496 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹) | |
2 | ndfatafv2nrn 43494 | . . . 4 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) | |
3 | df-nel 3123 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
4 | 2, 3 | sylib 220 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹) |
5 | 4 | con4i 114 | . 2 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐹 defAt 𝐴) |
6 | 1, 5 | impbii 211 | 1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2113 ∉ wnel 3122 ran crn 5549 defAt wdfat 43389 ''''cafv2 43481 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-dfat 43392 df-afv2 43482 |
This theorem is referenced by: dmafv2rnb 43502 afv2elrn 43504 tz6.12i-afv2 43516 afv2ndeffv0 43533 afv2rnfveq 43535 |
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