Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dffunsALTV5 | Structured version Visualization version GIF version |
Description: Alternate definition of the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021.) |
Ref | Expression |
---|---|
dffunsALTV5 | ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffunsALTV4 36570 | . 2 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥} | |
2 | ineccnvmo2 36265 | . . 3 ⊢ (∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑓𝑥) | |
3 | 2 | rabbii 3398 | . 2 ⊢ {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} = {𝑓 ∈ Rels ∣ ∀𝑢∃*𝑥 𝑢𝑓𝑥} |
4 | 1, 3 | eqtr4i 2770 | 1 ⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑥 ∈ ran 𝑓∀𝑦 ∈ ran 𝑓(𝑥 = 𝑦 ∨ ([𝑥]◡𝑓 ∩ [𝑦]◡𝑓) = ∅)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 ∀wal 1541 = wceq 1543 ∃*wmo 2539 ∀wral 3064 {crab 3068 ∩ cin 3882 ∅c0 4253 class class class wbr 5069 ◡ccnv 5567 ran crn 5569 [cec 8412 Rels crels 36108 FunsALTV cfunsALTV 36136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ral 3069 df-rex 3070 df-rmo 3072 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-ec 8416 df-coss 36310 df-rels 36376 df-ssr 36389 df-cnvrefs 36414 df-cnvrefrels 36415 df-funss 36564 df-funsALTV 36565 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |