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Mathbox for Alexander van der Vekens |
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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 45356 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹) | |
2 | ndfatafv2nrn 45354 | . . . 4 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) | |
3 | df-nel 3048 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹) |
5 | 4 | con4i 114 | . 2 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐹 defAt 𝐴) |
6 | 1, 5 | impbii 208 | 1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2106 ∉ wnel 3047 ran crn 5632 defAt wdfat 45249 ''''cafv2 45341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6445 df-fun 6495 df-dfat 45252 df-afv2 45342 |
This theorem is referenced by: dmafv2rnb 45362 afv2elrn 45364 tz6.12i-afv2 45376 afv2ndeffv0 45393 afv2rnfveq 45395 |
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