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Theorem uniimaelsetpreimafv 44411
Description: The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
uniimaelsetpreimafv ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑃
Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem uniimaelsetpreimafv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
210nelsetpreimafv 44405 . . . 4 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3 elnelne2 3049 . . . . . 6 ((𝑆𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4 n0 4235 . . . . . 6 (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦𝑆)
53, 4sylib 221 . . . . 5 ((𝑆𝑃 ∧ ∅ ∉ 𝑃) → ∃𝑦 𝑦𝑆)
65expcom 417 . . . 4 (∅ ∉ 𝑃 → (𝑆𝑃 → ∃𝑦 𝑦𝑆))
72, 6syl 17 . . 3 (𝐹 Fn 𝐴 → (𝑆𝑃 → ∃𝑦 𝑦𝑆))
87imp 410 . 2 ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑦 𝑦𝑆)
91imaelsetpreimafv 44410 . . . . . 6 ((𝐹 Fn 𝐴𝑆𝑃𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1093expa 1119 . . . . 5 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1110unieqd 4810 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
12 fvex 6689 . . . . 5 (𝐹𝑦) ∈ V
1312unisn 4818 . . . 4 {(𝐹𝑦)} = (𝐹𝑦)
1411, 13eqtrdi 2789 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = (𝐹𝑦))
15 dffn3 6517 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1615biimpi 219 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1716ad2antrr 726 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝐹:𝐴⟶ran 𝐹)
181elsetpreimafvssdm 44401 . . . . 5 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
1918sselda 3877 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝑦𝐴)
2017, 19ffvelrnd 6864 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ran 𝐹)
2114, 20eqeltrd 2833 . 2 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) ∈ ran 𝐹)
228, 21exlimddv 1942 1 ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wne 2934  wnel 3038  wrex 3054  c0 4211  {csn 4516   cuni 4796  ccnv 5524  ran crn 5526  cima 5528   Fn wfn 6334  wf 6335  cfv 6339
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 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347
This theorem is referenced by:  imasetpreimafvbijlemf  44416  fundcmpsurbijinjpreimafv  44422
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